MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equsexhv Structured version   Visualization version   GIF version

Theorem equsexhv 2162
Description: Version of equsexh 2331 with a dv condition, which does not require ax-13 2282. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
equsalhw.1 (𝜓 → ∀𝑥𝜓)
equsalhw.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsexhv (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem equsexhv
StepHypRef Expression
1 equsalhw.1 . . 3 (𝜓 → ∀𝑥𝜓)
21nf5i 2064 . 2 𝑥𝜓
3 equsalhw.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsexv 2147 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  cleljustALT  2221
  Copyright terms: Public domain W3C validator